LOS ANGELES—Lawyers for President Donald Trump’s personal attorney Michael Cohen asked a federal judge here Friday to halt a civil case against him in light of the criminal investigation embroiling him in New York.

“We’re asking for a brief hit-the-pause button,” said Brent Blakely, an attorney for Mr. Cohen, referring to a suit filed in early March by Stephanie Clifford, a former adult film actress. Ms. Clifford, known professionally as Stormy Daniels, sued to invalidate a 2016 agreement which, in exchange for a payment of...